A New Approach to Approximate Solutions for Nonlinear Differential Equation

Abstract

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

1. Introduction

In the last few years, the study of approximations methods for systems of differential equations has been extensively developed; see, for example, [1]. This technique, known as the perturbation method (see [2]), has many applications in the theory of fractional differentiation operators (see [3]), in reaction-diffusion equations, stochastic stability, and asymptotic stability (see [4–9]), and for some numerical considerations (see, for example, [10–12]).

In current applications, some considerations require only the use of a small number of terms in the perturbation expansion, but the simple application of the perturbation is problematic if we want to calculate a uniformly valid solution.

Therefore, to structure a uniformly valid solution, one must look for an approximation that eliminates the terms causing the problem (secular terms). A technique to avoid the presence of these terms has been developed by Lindstedt. The principle of the Lindstedt method is to find approximations for periodic solutions, by convergent series using the expansion theorem and the periodicity of the solution [13, 14]. This method has various applications and properties; see, for example, [15]. Later, Poincaré proved that the expansion obtained by the Lindstedt technique is both asymptotic and uniformly valid.

The aim of this work is to present an analytical approximation study of periodic solutions for systems of second-order nonlinear differential equations. Although our analysis is based on the Lindstedt method, nevertheless the chosen development is according to a different approach from the one usually used. Thus, we recover an improvement in the process of the approximation.

Our paper consists of three sections. In the first section we present the general framework of our study. In the second, we recall most of the preliminary notions and the necessary definitions, and we prove the third approximation in the general case. Finally, in Section 3 we define and study the approximations of a new nonclassical equation.

2. Preliminaries and Definitions

In this section, we present an approximation method, based on the expansion of a solution of a differential equation in a series in a small parameter. It is used to construct uniformly valid periodic solutions to second-order nonlinear differential equations in the formwith , where means that the positive parameter is small enough to be close to zero and is supposed to be an analytical function of and .

If we obtain the following nonperturbed problem:

Before we discuss our subject, we present some basic concepts concerning the perturbation theory. Then we introduce the Lindstedt method, which we use to determine uniformly valid solutions, in order to find a closer approximate solution for (2) ( is closer to than means that ). For further developments concerning the Lindstedt method see [16, 17].

2.1. Approximation Technique

We assume that the th approximate solution of (1) can be written asThe general procedure of the simple approximation is to substitute (3) into (1), develop in powers of , and put all coefficients of the powers of equal to zero. This gives a system of linear nonhomogeneous differential equations that we can solve recursively.

But the simple approximation takes us on a problem, if we need to calculate an analytical approximations of periodic solutions of nonlinear differential equations in the form given by (1). We illustrate this type of difficulty in the following example.

2.1.1. Example

We apply the simple approximation to the following equation:with the initial values .

The fourth approximate solution of (4) is . After substituting and calculating, we find

We remark that the terms and are nonperiodic and unbounded as . This leads to the notion of secular terms.

2.2. Secular Terms

The conservation of a finite numbers of terms on the right-side of expansion (5) determines a function that is not only nonperiodic, but also unbounded as .

Definition 1. Terms such as or where are called secular terms.

These expressions appear because expansion (5) is not uniformly valid. The existence of such expressions destroys the periodicity of expansion (5) when only a finite number of terms is conserved. Therefore, to obtain a uniformly valid solution, we must look for an approximation that eliminates secular terms. A technique to avoid the presence of secular terms and allows for an approximation that is valid for all time has been developed by Lindstedt-Poincaré as described above in what follows.

2.3. Lindstedt-Poincaré Method

The substance of this method is to introduce a new independent variable linearly linked to the old independent variable. This transformation completely eliminates the secular terms. The basic idea came from the astronomer Lindstedt, based on the change of variable with , and both and are expanded in powers of as follows:and we note that, in this step, are unknowns; we obtain them by elimination of the secular terms.

First, we introduce the following notations:and (1) becomeswith . When we substitute expansion (6) into (8) we have Then we put all different powers of to zero, and we obtain (10), (11), (12), (13), and (15), such thatnote here that , is also an analytical function of .

To calculate an approximate periodic solutions of (8), we must solve (11), (12), (13), and (15). The following proposition gives the general formula of periodic solutions. Although these results are in [17], they are not detailed.

Proposition 2. We consider the following equation:and the solution of problem (16) isMoreover, problem (16) has a periodic solution if and only if

Proof. We know that the solution of (16) is such that . By variation of constants we find .
with the initial values ; we have , so we deduce that problem (16) admits (17) as a solution.
Moreover, (16) gives On the other hand, the condition of periodicity for the new variable can be expressed as , so the corresponding conditions for are which yields to the periodicity condition ,According to (11) we have ; we rewrite (22) as which is required.

2.3.1. Example

We apply the method of Lindstedt to (4) with the initial values , and we calculate according to Proposition 2. Thus, we find that the fourth approximation of the periodic solution of (4) iswith such that

Remark 3. Although the calculation of is very long, usually in applications, the fourth approximation is among the high orders that are often useful. For this reason, we give its equation in the next proposition.

Remark 4. The Lindstedt method gives only periodic solutions.

3. Our Results

3.1. General Formula

Practically, for many considerations we are forced to use a small number of terms in the perturbation expansion. We note here that the second and third terms are determined by (11) and (12) in [17]. In the following proposition, (13) which determines the fourth term is explicitly stated.

Proposition 5. The general formula of (13) is

Proof. First, (9) gives such that with On the other hand, in the third order we have and and also and when we substitute (31) into (26), we get (25).

3.2. Main Result

In this essential part of our work, we deal with some nonclassical equations, more general than (1), and also different from the equation studied in [2]. We consider equations in the following form:with , where is a small positive parameter and is supposed to be an analytical function of and .

To compute an uniformly approximate periodic solution, a new variable is introduced, and both and are expanded in powers of as follows:withWe note that, in this step, are unknowns, and we obtain them by elimination of the secular terms.

To use the uniformly approximate periodic solution (33), we give firstly the general formula of , and in the following proposition.

Proposition 6. The terms are, respectively, solutions of (35), (36), and (37) such thatwith ,

Proof. Equation (32) will be written as where is an analytical function of and .
Substituting (33) into (32), we have then we put all different powers of to zero, and we obtain (35), (36), (37), and so on.

The aim of this study is to construct a new approach to (2), which gives a closer approximate solution of (2) more than an approximate solution of (1). The relations between an approximate solution of (32) and that of (1) are determined by the following lemma.

Lemma 7. If the function is expanded in powers of with , where are real constants, we have
.
.
.

Proof. Equation (35) gives .
When we apply the periodicity condition (22) to (36), we have On the other hand, according to (17) the solution of (36) is given by When we apply the periodicity condition (22) to (12), we haveOn the other hand, according to (17) the solution of (11) is given by

Theorem 8. If the function is expanded in powers of with , where are real constants, so one has the following:
(1) If , the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1).
(2) If and , the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1).
Moreover, the approximate solutions of (32) are closer to the solutions of (2) more than the approximate solutions of (1) where .

Proof. According to the results given by Lemma 7, we have the following:
If and we have So is an approximation to closer than .
If , we haveSince , for small enough, there exists a positive real constant such that .
Let be a function defined by . So, is continuous with , then , we have .
Therefore, for all we getwhich implies from what precedes that . So is an approximation to closer than .
Moreover, if , then such that, ,So is an approximation to closer than .

Remark 9. Let be a real function such that , where can be expanded in powers of , with real constants.
If or 2, according to the conditions of Theorem 8, we conclude that is an approximation to closer than .
If (in the case where , the fourth term is given by Proposition 5), we can expect that is an approximation to closer than .

Remark 10. We note here that, in the fractional case, the existence of a positive solution of (32) is studied in [18].

Remark 11. Although the Lindstedt-Poincaré method gives uniformly valid asymptotic expansions for periodic solutions of weakly nonlinear oscillations, i.e., , the technique does not work if the amplitude of the oscillation is a function of time (see [16, 17]).

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.